Integrate and Classify Normal Distributions · Getting started
Tools to integrate normal (Gaussian) distributions in any dimensions with any parameters within any domain, compute pdf/cdf/inverse cdf of any function of a normal vector, and compute quantities concerning classification performance among two or more multinormals, such as error matrix and discriminability 
.
.For function help, type:
doc integrate_normal
doc classify_normals
doc classify_normals_multi
doc norm_fun_cdf
doc norm_fun_pdf
doc norm_fun_inv
If you use this toolbox, please cite: A method to integrate and classify normal distributions. Some examples marked here are described in the paper.
Bugs/questions/comments to abhranil.das@utexas.edu.
Abhranil Das, Center for Perceptual Systems, University of Texas at Austin
Integrate normal in 1d
Define normal parameters, and a quadratic function of it, 
, i.e. q(x)= -1*x^2 + 1*x + 1:
, i.e. q(x)= -1*x^2 + 1*x + 1:mu=-2; v=3;
quad.q2=-1;
quad.q1=1;
quad.q0=1;
Integrate the normal probability in the quadratic domain defined by 
:
p=integrate_normal(mu,v,quad)
p = 0.1941
Most plots can be zoomed and panned.
pdf/cdf/inverse cdf of 
p=norm_fun_cdf([0 1 2],mu,v,quad)
p = 1×3
0.8059 0.9175 1.0000
f=norm_fun_pdf([0 1 2],mu,v,quad)
f = 1×3
0.0865 0.1696 0
fplot(@(x) norm_fun_pdf(x,mu,v,quad),[-30 5]); ylabel 'pdf'
% 1st, 2nd and 3rd quartiles
x=norm_fun_inv([0.25 0.5 0.75],mu,v,quad)
x = 1×3
-12.2134 -5.0417 -0.7030
normal probability in the domain defined by a non-quadratic function 
fun=@(x) cos(x.^2);
[p,pc,bd_pts]=integrate_normal(mu,v,fun,'dom_type','fun','fun_span',5,'fun_resol',500)
p = 0.5803
pc = 0.4197
bd_pts = 1×100
-10.5606 -10.4108 -10.2588 -10.1045 -9.9479 -9.7887 -9.6269 -9.4623 -9.2948 -9.1243 -8.9505 -8.7732 -8.5923 -8.4075 -8.2185 -8.0251 -7.8269 -7.6236 -7.4147 -7.1997 -6.9782 -6.7493 -6.5124 -6.2666 -6.0107 -5.7434 -5.4631 -5.1675 -4.8541 -4.5189 -4.1568 -3.7599 -3.3160 -2.8025 -2.1708 -1.2533 1.2533 2.1708 2.8025 3.3160 3.7599 4.1568 4.5189 4.8541 5.1675 5.4631 5.7434 6.0107 6.2666 6.5124
pdf of 
x=linspace(-2,2,100);
f=norm_fun_pdf(x,mu,v,fun,'fun_span',5,'fun_resol',500,'dx',1e-2);
plot(x,f); ylabel 'pdf'
cdf of
F=norm_fun_cdf(x,mu,v,fun,'fun_span',5,'fun_resol',500);
plot(x,F); ylabel 'cdf'
Inverse cdf of
p=[0.5 0.75]; % median and 3rd quartile
x=norm_fun_inv(p,mu,v,fun)
x = 1×2
0.2803 0.8832
Classify in 1d
mu_1=0; v_1=1;
mu_2=2.5; v_2=1.5;
results=classify_normals([mu_1,v_1],[mu_2,v_2])
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [-11.2223 1.2223]
norm_errmat: [2×2 double]
norm_err: 0.1296
norm_dprime: 2.2565
norm_maha_dprime: 2.2361
with unequal priors and outcome values
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[3 0; 0 1])
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [-12.0987 2.0987]
norm_errmat: [2×2 double]
norm_err: 0.1240
norm_valmat: [2×2 double]
norm_val: 2.2509
Classify in 1d using samples
Priors are assumed prop. to sample sizes.
mu_1=0; v_1=1;
samp_1=normrnd(mu_1,sqrt(v_1),[700 1]);
mu_2=1.5; v_2=1.5;
samp_2=normrnd(mu_2,sqrt(v_2),[300 1]);
results=classify_normals(samp_1,samp_2,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -813
1 4 -816 initial simplex
2 6 -817 reflect
3 8 -818 reflect
4 13 -818 shrink
5 14 -818 reflect
6 16 -818 contract inside
7 18 -818 contract inside
8 20 -818 contract inside
9 25 -818 shrink
10 30 -818 shrink
11 35 -818 shrink
12 40 -818 shrink
13 45 -818 shrink
14 50 -818 shrink
15 55 -818 shrink
16 60 -818 shrink
17 65 -818 shrink
18 70 -818 shrink
19 75 -818 shrink
20 80 -818 shrink
21 85 -818 shrink
22 90 -818 shrink
23 95 -818 shrink
24 100 -818 shrink
25 105 -818 shrink
26 110 -818 shrink
27 115 -818 shrink
28 120 -818 shrink
29 125 -818 shrink
30 130 -818 shrink
31 135 -818 shrink
32 140 -818 shrink
33 145 -818 shrink
34 150 -818 shrink
35 155 -818 shrink
36 160 -818 shrink
37 165 -818 shrink
38 170 -818 shrink
39 175 -818 shrink
40 180 -818 shrink
41 185 -818 shrink
42 190 -818 shrink
43 195 -818 shrink
44 200 -818 shrink
45 205 -818 shrink
46 210 -818 shrink
47 215 -818 shrink
48 220 -818 shrink
49 225 -818 shrink
50 230 -818 shrink
51 235 -818 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-03
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [-8.4926 1.4010]
norm_errmat: [2×2 double]
norm_err: 0.1888
samp_errmat: [2×2 double]
samp_err: 0.1870
samp_dv: {[700×1 double] [300×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[700×1 double] [300×1 double]}
samp_opt_bd_pts: [-9.0121 1.3127]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.1820
Integrate normal in 2d
in a quadratic domain
defined by 
, i.e.
, i.e.q(x,y)= [x y]*[1 1; 1 1]*[x;y] + [-1 0]*[x;y] -1 > 0
mu=[-1; -1]; v=[1 0.5; 0.5 2];
quad.q2=[1 1; 1 1];
quad.q1=[-1;0];
quad.q0=-1;
% compare two integration algorithms
[p,pc,bd_pts]=integrate_normal(mu,v,quad) % ray method
p = 0.8214
pc = 0.1786
bd_pts = 2×504
103 ×
-0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0021 -0.0001 0.0020 -0.0001 0.0020 -0.0001 0.0019 -0.0002 0.0019 -0.0002 0.0019 -0.0002 0.0018 -0.0002 0.0018 -0.0002 0.0018 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0017 -0.0002 0.0016 -0.0002 0.0015 -0.0002 0.0015 -0.0002 0.0014 -0.0002 0.0013 -0.0003 0.0013
-0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0003 -0.0008 -0.0002 -0.0008 -0.0002 -0.0008 -0.0002 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0001 -0.0007 -0.0000 -0.0007 0.0000 -0.0007 0.0000 -0.0007 0.0001 -0.0006 0.0001 -0.0006 0.0002 -0.0006 0.0002
axis([-6 10 -10 2])
[p,pc]=integrate_normal(mu,v,quad,'method','gx2') % gx2 method
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.2e-03. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.2e-03. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
p = 0.8223
pc = 0.1777
axis([-6 10 -10 2])
[paper fig. 2c] In a non-quadratic domain
defined by 
:
:fun=@(x,y) x.*sin(y) - y.*cos(x) -1;
integrate_normal(mu,v,fun,'dom_type','fun','fun_span',3);
axis([-5 5 -5 5])
pdf of 
x=linspace(-10,10,100);
plot(x,norm_fun_pdf(x,mu,v,fun,'dx',1e-1))
[paper sec. 2.2.5] probabilities of a function 
of a 2d normal
mu=[1;2]; v=.01*[1 1.6; 1.6 4];
fun=@(x,y) x.^y;
cdf
dx=.01; x=0:dx:3;
F=norm_fun_cdf(x,mu,v,fun,'fun_span',5,'fun_resol',500);
plot(x,F); xlabel x^y; ylabel 'cdf'
f=norm_fun_pdf(x,mu,v,fun,'fun_span',5,'fun_resol',500,'dx',1e-2);
plot(x,f); xlabel x^y; ylabel 'pdf'
Verify that pdf sums to 1:
sum(f)*dx
ans = 1.0000
Mean of 
m=sum(x.*f)*dx
m = 1.0266
sd of
m2=sum(x.^2.*f)*dx;
sd=sqrt(m2-m^2)
sd = 0.2115
median of
norm_fun_inv(.5,mu,v,fun)
ans = 1.0000
Joint probability of functions of a normal
% (x,y) is a normal vector with these parameters:
mu=[6;-19];
v=[1 -.7; -.7 2];
% functions of the normal vector
fun1=@(x,y) cos(4*x);
fun2=@(x,y) cos(4*y);
% p(cos 4x > 0)
integrate_normal(mu,v,fun1,'dom_type','fun','fun_span',3);
axis image; axis([2 10 -27 -11])
% p(cos 4y > 0)
integrate_normal(mu,v,fun2,'dom_type','fun','fun_span',3);
axis image; axis([2 10 -27 -11])
Probability that 
and 
are both > 0, i.e. integrate the normal in the domain 
:
and are both > 0, i.e. integrate the normal in the domain :fun3=@(x,y) min(fun1(x,y),fun2(x,y));
p=integrate_normal(mu,v,fun3,'dom_type','fun','fun_span',3)
p = 0.2495
axis image; axis([2 10 -27 -11])
Probability that 
, i.e. integrate the normal in the domain 
, i.e. integrate the normal in the domain fun4=@(x,y) fun1(x,y)+fun2(x,y)-0.5;
p=integrate_normal(mu,v,fun4,'dom_type','fun','fun_span',3)
p = 0.3072
axis image; axis([2 10 -27 -11]);
Percentage points (inverse cdf):
p=[0.5 0.75];
x=norm_fun_inv(p,mu,v,fun4)
x = 1×2
-0.4999 0.2210
Classify two 2d normals, one inside the other
mu_1=[4; 5]; v_1=[2 1; 1 1];
mu_2=mu_1; v_2=3*[2 -1; -1 1];
% compare two integration algorithms
results_ray=classify_normals([mu_1,v_1],[mu_2,v_2]) % ray method
results_ray = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [2×414 double]
norm_errmat: [2×2 double]
norm_err: 0.1979
norm_dprime: 1.6982
norm_maha_dprime: 0
axis([-2 8 0 10])
results_gx2=classify_normals([mu_1,v_1],[mu_2,v_2],'method','gx2') % gx2 method
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.7e-06. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.7e-06. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
results_gx2 = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1979
norm_dprime: 1.6982
norm_maha_dprime: 0
axis([-2 8 0 10])
[paper fig. 2d] Classify in 2d with custom boundaries and from samples
mu_1=[2;4]; v_1=[1 1.5; 1.5 3];
mu_2=[5;0]; v_2=[3 0; 0 1];
% ray method
results_ray=classify_normals([mu_1,v_1],[mu_2,v_2]);
axis image; axis([-10 10 -10 10])
results_ray.norm_err
ans = 0.0079
% compare with generalized chi square method
results_gx2=classify_normals([mu_1,v_1],[mu_2,v_2],'method','gx2');
axis image; axis([-10 10 -10 10])
results_gx2.norm_err
ans = 0.0079
% now supply a custom linear boundary
linear_bd.q2=zeros(2);
linear_bd.q1=[-.7;1];
linear_bd.q0=0;
results_linear=classify_normals([mu_1,v_1],[mu_2,v_2],'dom',linear_bd)
results_linear = struct with fields:
norm_bd_pts: [2×300 double]
norm_errmat: [2×2 double]
norm_err: 0.0133
axis image; axis([-10 10 -10 10])
% now classify using samples
n_samp=1e3;
samp_1=mvnrnd(mu_1,v_1,n_samp);
samp_2=mvnrnd(mu_2,v_2,n_samp);
results_samp=classify_normals(samp_1,samp_2,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -1987
1 7 -1987 initial simplex
2 8 -1987 reflect
3 16 -1987 shrink
4 18 -1987 contract outside
5 26 -1987 shrink
6 34 -1987 shrink
7 42 -1987 shrink
8 50 -1987 shrink
9 58 -1987 shrink
10 66 -1987 shrink
11 74 -1987 shrink
12 82 -1987 shrink
13 90 -1987 shrink
14 98 -1987 shrink
15 106 -1987 shrink
16 114 -1987 shrink
17 122 -1987 shrink
18 130 -1987 shrink
19 138 -1987 shrink
20 146 -1987 shrink
21 154 -1987 shrink
22 162 -1987 shrink
23 170 -1987 shrink
24 178 -1987 shrink
25 186 -1987 shrink
26 194 -1987 shrink
27 202 -1987 shrink
28 210 -1987 shrink
29 218 -1987 shrink
30 226 -1987 shrink
31 234 -1987 shrink
32 242 -1987 shrink
33 250 -1987 shrink
34 258 -1987 shrink
35 266 -1987 shrink
36 274 -1987 shrink
37 282 -1987 shrink
38 290 -1987 shrink
39 298 -1987 shrink
40 306 -1987 shrink
41 314 -1987 shrink
42 322 -1987 shrink
43 330 -1987 shrink
44 338 -1987 shrink
45 346 -1987 shrink
46 354 -1987 shrink
47 362 -1987 shrink
48 370 -1987 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results_samp = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [2×516 double]
norm_errmat: [2×2 double]
norm_err: 0.0085
norm_dprime: 4.7736
norm_maha_dprime: 4.3714
samp_errmat: [2×2 double]
samp_err: 0.0065
samp_dprime: 4.9675
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd_pts: [2×32450 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.0065
samp_opt_dprime: 4.9675
axis image; axis([-10 10 -10 10])
% modify the sample-optimized boundary and classify with it
custom_bd=results_samp.samp_opt_bd;
custom_bd.q2=custom_bd.q2+.1;
custom_bd.q1=custom_bd.q1-2.5;
custom_bd.q0=custom_bd.q0+5;
results_samp_custom=classify_normals(samp_1,samp_2,'input_type','samp','dom',custom_bd)
Iteration Func-count min f(x) Procedure
0 1 -1942
1 7 -1964 initial simplex
2 8 -1964 reflect
3 10 -1976 expand
4 11 -1976 reflect
5 12 -1976 reflect
6 14 -1981 expand
7 15 -1981 reflect
8 16 -1981 reflect
9 17 -1981 reflect
10 19 -1983 reflect
11 20 -1983 reflect
12 21 -1983 reflect
13 22 -1983 reflect
14 24 -1983 contract inside
15 26 -1983 contract inside
16 28 -1983 contract inside
17 30 -1983 contract inside
18 31 -1983 reflect
19 39 -1983 shrink
20 41 -1983 contract outside
21 49 -1983 shrink
22 50 -1983 reflect
23 51 -1983 reflect
24 52 -1983 reflect
25 54 -1983 contract inside
26 56 -1983 contract inside
27 64 -1983 shrink
28 72 -1983 shrink
29 80 -1983 shrink
30 88 -1983 shrink
31 96 -1983 shrink
32 104 -1983 shrink
33 112 -1983 shrink
34 120 -1983 shrink
35 128 -1983 shrink
36 136 -1983 shrink
37 144 -1983 shrink
38 152 -1983 shrink
39 160 -1983 shrink
40 168 -1983 shrink
41 176 -1983 shrink
42 184 -1983 shrink
43 192 -1983 shrink
44 200 -1983 shrink
45 208 -1983 shrink
46 216 -1983 shrink
47 224 -1983 shrink
48 232 -1983 shrink
49 240 -1983 shrink
50 248 -1983 shrink
51 256 -1983 shrink
52 264 -1983 shrink
53 272 -1983 shrink
54 280 -1983 shrink
55 288 -1983 shrink
56 296 -1983 shrink
57 304 -1983 shrink
58 312 -1983 shrink
59 320 -1983 shrink
60 328 -1983 shrink
61 336 -1983 shrink
62 344 -1983 shrink
63 352 -1983 shrink
64 360 -1983 shrink
65 368 -1983 shrink
66 376 -1983 shrink
67 384 -1983 shrink
68 392 -1983 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results_samp_custom = struct with fields:
norm_bd_pts: [2×438 double]
norm_errmat: [2×2 double]
norm_err: 0.0275
samp_errmat: [2×2 double]
samp_err: 0.0290
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd_pts: [2×26558 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.0085
axis image; axis([-10 10 -10 10])
[paper fig. 2e] Classify in 2d with non-normal samples
n_samp=1e3;
samp_1=exp(mvnrnd([0 0],eye(2),n_samp));
samp_2=-exp(mvnrnd([1 1],eye(2),n_samp))+[2 2];
results=classify_normals(samp_1,samp_2,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -1485
1 7 -1490 initial simplex
2 9 -1507 expand
3 10 -1507 reflect
4 11 -1507 reflect
5 12 -1507 reflect
6 13 -1507 reflect
7 15 -1522 expand
8 16 -1522 reflect
9 18 -1540 expand
10 19 -1540 reflect
11 21 -1563 expand
12 22 -1563 reflect
13 24 -1594 expand
14 25 -1594 reflect
15 26 -1594 reflect
16 28 -1642 expand
17 29 -1642 reflect
18 31 -1719 expand
19 32 -1719 reflect
20 34 -1724 reflect
21 35 -1724 reflect
22 37 -1724 contract inside
23 38 -1724 reflect
24 40 -1724 contract inside
25 42 -1724 contract inside
26 43 -1724 reflect
27 45 -1724 contract inside
28 47 -1724 contract inside
29 49 -1724 contract inside
30 51 -1724 contract inside
31 53 -1724 contract inside
32 55 -1724 contract inside
33 56 -1724 reflect
34 58 -1724 contract inside
35 60 -1726 contract inside
36 62 -1726 contract inside
37 64 -1726 contract outside
38 65 -1726 reflect
39 67 -1726 contract outside
40 69 -1726 contract inside
41 77 -1727 shrink
42 78 -1727 reflect
43 80 -1727 contract inside
44 88 -1728 shrink
45 89 -1728 reflect
46 90 -1728 reflect
47 92 -1728 contract inside
48 93 -1728 reflect
49 95 -1729 reflect
50 96 -1729 reflect
51 97 -1729 reflect
52 105 -1729 shrink
53 107 -1729 contract outside
54 109 -1729 contract outside
55 117 -1729 shrink
56 118 -1729 reflect
57 119 -1729 reflect
58 121 -1729 contract outside
59 129 -1729 shrink
60 137 -1729 shrink
61 145 -1729 shrink
62 153 -1729 shrink
63 161 -1729 shrink
64 169 -1729 shrink
65 177 -1729 shrink
66 185 -1729 shrink
67 193 -1729 shrink
68 201 -1729 shrink
69 209 -1729 shrink
70 217 -1729 shrink
71 225 -1729 shrink
72 233 -1729 shrink
73 241 -1729 shrink
74 249 -1729 shrink
75 257 -1729 shrink
76 265 -1729 shrink
77 273 -1729 shrink
78 281 -1729 shrink
79 289 -1729 shrink
80 297 -1729 shrink
81 305 -1729 shrink
82 313 -1729 shrink
83 321 -1729 shrink
84 329 -1729 shrink
85 337 -1729 shrink
86 345 -1729 shrink
87 353 -1729 shrink
88 361 -1729 shrink
89 369 -1729 shrink
90 377 -1729 shrink
91 385 -1729 shrink
92 393 -1729 shrink
93 401 -1729 shrink
94 409 -1729 shrink
95 417 -1729 shrink
96 425 -1729 shrink
97 433 -1729 shrink
98 441 -1729 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [2×514 double]
norm_errmat: [2×2 double]
norm_err: 0.1425
norm_dprime: 2.1384
norm_maha_dprime: 1.3361
samp_errmat: [2×2 double]
samp_err: 0.2575
samp_dprime: 1.3021
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd_pts: [2×27770 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.1355
samp_opt_dprime: 2.2015
axis image; axis([-20 15 -20 15])
Inversion/union/intersection of integration/classification domains
mu=[0;0]; v=[.5 0; 0 1];
circle_left=@(x,y) -(x+1).^2-y.^2+5;
circle_right=@(x,y) -(x-1).^2-y.^2+5;
circle_union=@(x,y) max(circle_left(x,y), circle_right(x,y));
p=integrate_normal(mu,v,circle_union,'dom_type','fun','fun_span',5)
p = 0.9680
axis image; axis([-4 4 -4 4])
circle_intersection=@(x,y) min(circle_left(x,y), circle_right(x,y));
p=integrate_normal(mu,v,circle_intersection,'dom_type','fun','fun_span',5)
p = 0.7949
axis image; axis([-4 4 -4 4])
crescent=@(x,y) max(circle_left(x,y), -circle_right(x,y));
p=integrate_normal(mu,v,crescent,'dom_type','fun','fun_span',5)
p = 0.9136
axis image; axis([-4 4 -4 4])
% classify normals using this crescent domain
mu_2=[2.2;0]; v_2=[.5 0; 0 .25];
results=classify_normals([mu,v],[mu_2,v_2],'dom',crescent,'dom_type','fun')
results = struct with fields:
norm_bd_pts: [2×586 double]
norm_errmat: [2×2 double]
norm_err: 0.1235
axis image; axis([-2 4 -3 3])
Classify in 3d
mu_1=[0;0;0]; v_1=eye(3);
mu_2=[2;1;1]; v_2=2*eye(3);
results=classify_normals([mu_1,v_1],[mu_2,v_2])
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [3×11936 double]
norm_errmat: [2×2 double]
norm_err: 0.1390
norm_dprime: 2.1694
norm_maha_dprime: 2
Estimate tiny errors (large 
) with high accuracy
format long
dprime_true=75
dprime_true =
75
mu_1=[0;0;0]; v=eye(3);
mu_2=dprime_true*[1;0;0];
results=classify_normals([mu_1,v],[mu_2,v],'AbsTol',0,'RelTol',1e-2)
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [3×14112 double]
norm_errmat: [2×2 double]
norm_err: 4.608155172364448e-308
norm_dprime: 74.999967581801229
norm_maha_dprime: 75
axis image; axis([-5 80 -2 2 -2 2]); view(-53,25)
dprime_computed=results.norm_dprime
dprime_computed =
74.999967581801229
format
[paper fig. 4a] accuracy vs separation between the normals
mu_1=[0;0;0];
% both normals have the same covariance, so we can check against
% the true d' (Mahalanobis distance).
v=[1 .5 .7;
.5 2 1 ;
.7 1 3];
steps=linspace(1,100,10);
d_true=nan(size(steps));
d_gx2=nan(size(steps));
d_ray=nan(size(steps));
parfor i=1:length(steps)
i
mu_2=steps(i)*[1;1;1];
results_gx2=classify_normals([mu_1,v],[mu_2,v],'method','gx2','AbsTol',0,'RelTol',0,'plotmode',false);
d_true(i)=results_gx2.norm_maha_dprime;
d_gx2(i)=results_gx2.norm_dprime;
results_ray=classify_normals([mu_1,v],[mu_2,v],'method','ray','AbsTol',0,'RelTol',0,'plotmode',false);
d_ray(i)=results_ray.norm_dprime;
end
Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 4).
ans =
1
ans =
2
ans =
3
ans =
4
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 45)
In integrate_normal (line 88)
In classify_normals (line 194)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 44)
In integrate_normal (line 88)
In classify_normals (line 196)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
ans =
10
ans =
9
ans =
8
ans =
7
ans =
6
ans =
5
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 45)
In integrate_normal (line 88)
In classify_normals (line 194)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
Warning: Integrand function outputs did not match to the required tolerance when the same input values were supplied in two separate calls with different size input matrices. Check that the function is vectorized properly.
> In integral2Calc>integral2t/tensor (line 247)
In integral2Calc>integral2t (line 55)
In integral2Calc (line 9)
In integral2 (line 106)
In int_norm_ray (line 44)
In integrate_normal (line 88)
In classify_normals (line 196)
In parallel_function>make_general_channel/channel_general (line 832)
In remoteParallelFunction (line 67)
rel_err_gx2=abs(d_gx2-d_true)./d_true;
rel_err_ray=abs(d_ray-d_true)./d_true;
figure; hold on
plot(d_true,rel_err_gx2,'-o')
plot(d_true,rel_err_ray,'-o')
xline(-2*norminv(realmin)) % largest computable d', corr. to the smallest possible error representable in double-precision
yline(eps) % machine epsilon for double precision
xlim([1 80]); ylim([0 2.2e-15])
xlabel 'true d'''
ylabel('$\epsilon$','interpreter','latex')
legend({'$\tilde{\chi}^2$','ray'},'interpreter','latex')
[paper fig. 2a] Integrate normal in a torus defined by function 
mu=[0;0;0];
v=[1 0 0;
0 8 4;
0 4 8];
% plot the error ellipsoid of the normal
figure; plot_normal(mu,v);
fun_torus=@(x1,x2,x3) 1.5-(5-(x1.^2+x2.^2).^0.5).^2-x3.^2;
integrate_normal(mu,v,fun_torus,'dom_type','fun','fun_span',3,'fun_resol',10,'RelTol',1e-1);
axis image; axis([-7 7 -7 7 -7 7])
Plot the pdf and cdf of
x=[linspace(-80,-20,30) linspace(-20,10,20)];
f=norm_fun_pdf(x,mu,v,fun_torus,'fun_span',3,'fun_resol',10,'RelTol',1e-1,'dx',8);
plot(x,f)
p=norm_fun_cdf(x,mu,v,fun_torus,'fun_span',3,'fun_resol',10,'RelTol',1e-1);
plot(x,p)
[paper fig. 2g, top] Integrate normal in 4d
outside the sphere of radius 5, i.e. 
, i.e. x'*eye(4)*x + zeros(4,1)'*x -25 >0
, i.e. x'*eye(4)*x + zeros(4,1)'*x -25 >0mu=[1;1;1;1]; v=diag([1 2 3 4]);
quad.q2=eye(4);
quad.q1=zeros(4,1);
quad.q0=-25;
p=integrate_normal(mu,v,quad)
p = 0.1276
axis([-30 40 0 .06]);
[paper fig. 2g, bottom] Classify in 4d, with priors and outcome values
mu_1=[0;0;0;0]; v_1=eye(4);
mu_2=[1;1;1;1]; v_2=eye(4);
% here the Bayes decision variable is normally distributed
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[2 0; 0 1])
results = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1496
norm_valmat: [2×2 double]
norm_val: 1.5236
mu_1=[0;0;0;0];
v_1=[1 0 0 0;
0 2 -1 0;
0 -1 3 2;
0 0 2 4];
mu_2=1.5*[1;1;1;1];
v_2=[2 0 0 0;
0 2 0 0;
0 0 2 0;
0 0 0 1];
% correct classification of class 1 is valued 4x than class 2
% here the Bayes decision variable is distributed as a generalized chi-square
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[4 0; 0 1])
results = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1302
norm_valmat: [2×2 double]
norm_val: 2.9431
% use ray method (with Monte Carlo integration)
results=classify_normals([mu_1,v_1],[mu_2,v_2],'prior_1',.7,'vals',[4 0; 0 1],'method','ray','mc_samples',1e3)
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [4×2436 double]
norm_errmat: [2×2 double]
norm_err: 0.1289
norm_valmat: [2×2 double]
norm_val: 2.9459
% now classify using samples
n_samp=1e3;
results=classify_normals(mvnrnd(mu_1',v_1,n_samp),mvnrnd(mu_2',v_2,n_samp),'input_type','samp','prior_1',.7,'vals',[4 0; 0 1])
Iteration Func-count min f(x) Procedure
0 1 -4506
1 16 -4512 initial simplex
2 18 -4537 expand
3 19 -4537 reflect
4 20 -4537 reflect
5 21 -4537 reflect
6 23 -4537 contract outside
7 24 -4537 reflect
8 25 -4537 reflect
9 26 -4537 reflect
10 27 -4537 reflect
11 28 -4537 reflect
12 29 -4537 reflect
13 30 -4537 reflect
14 31 -4537 reflect
15 32 -4537 reflect
16 33 -4537 reflect
17 34 -4537 reflect
18 35 -4537 reflect
19 36 -4537 reflect
20 37 -4537 reflect
21 38 -4537 reflect
22 39 -4537 reflect
23 40 -4537 reflect
24 41 -4537 reflect
25 42 -4537 reflect
26 44 -4550 expand
27 45 -4550 reflect
28 46 -4550 reflect
29 47 -4550 reflect
30 48 -4550 reflect
31 49 -4550 reflect
32 50 -4550 reflect
33 51 -4550 reflect
34 52 -4550 reflect
35 53 -4550 reflect
36 54 -4550 reflect
37 55 -4550 reflect
38 56 -4550 reflect
39 57 -4550 reflect
40 58 -4550 reflect
41 59 -4550 reflect
42 61 -4551 reflect
43 62 -4551 reflect
44 79 -4553 shrink
45 80 -4553 reflect
46 81 -4553 reflect
47 82 -4553 reflect
48 84 -4553 contract outside
49 85 -4553 reflect
50 86 -4553 reflect
51 87 -4553 reflect
52 88 -4553 reflect
53 90 -4553 contract inside
54 92 -4553 contract inside
55 93 -4553 reflect
56 110 -4553 shrink
57 112 -4553 contract outside
58 113 -4553 reflect
59 114 -4553 reflect
60 115 -4553 reflect
61 116 -4553 reflect
62 117 -4553 reflect
63 118 -4553 reflect
64 119 -4553 reflect
65 120 -4553 reflect
66 121 -4553 reflect
67 123 -4554 reflect
68 125 -4554 contract inside
69 126 -4554 reflect
70 143 -4554 shrink
71 144 -4554 reflect
72 145 -4554 reflect
73 146 -4554 reflect
74 148 -4554 contract outside
75 150 -4554 contract outside
76 152 -4554 contract inside
77 153 -4554 reflect
78 154 -4554 reflect
79 155 -4554 reflect
80 156 -4554 reflect
81 173 -4554 shrink
82 190 -4554 shrink
83 207 -4554 shrink
84 224 -4554 shrink
85 225 -4554 reflect
86 226 -4554 reflect
87 227 -4554 reflect
88 228 -4554 reflect
89 229 -4554 reflect
90 230 -4554 reflect
91 232 -4554 contract outside
92 249 -4554 shrink
93 266 -4554 shrink
94 283 -4554 shrink
95 300 -4554 shrink
96 317 -4554 shrink
97 334 -4554 shrink
98 351 -4554 shrink
99 368 -4554 shrink
100 385 -4554 shrink
101 402 -4554 shrink
102 419 -4554 shrink
103 436 -4554 shrink
104 453 -4554 shrink
105 470 -4554 shrink
106 487 -4554 shrink
107 504 -4554 shrink
108 521 -4554 shrink
109 538 -4554 shrink
110 555 -4554 shrink
111 572 -4554 shrink
112 589 -4554 shrink
113 606 -4554 shrink
114 623 -4554 shrink
115 640 -4554 shrink
116 657 -4554 shrink
117 674 -4554 shrink
118 691 -4554 shrink
119 708 -4554 shrink
120 725 -4554 shrink
121 742 -4554 shrink
122 759 -4554 shrink
123 776 -4554 shrink
124 793 -4554 shrink
125 810 -4554 shrink
126 827 -4554 shrink
127 844 -4554 shrink
128 861 -4554 shrink
129 878 -4554 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-04
results = struct with fields:
norm_bd: [1×1 struct]
norm_errmat: [2×2 double]
norm_err: 0.1328
norm_valmat: [2×2 double]
norm_val: 2.9402
samp_errmat: [2×2 double]
samp_err: 0.2170
samp_valmat: [2×2 double]
samp_val: 4506
samp_dv: {[1000×1 double] [1000×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[1000×1 double] [1000×1 double]}
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.1570
samp_opt_valmat: [2×2 double]
samp_opt_val: 4554
[paper fig. 2b] Integrate in a 4d polyhedron, using Monte Carlo
mu=zeros(4,1);
v=eye(4);
fun=@(x1,x2,x3,x4) abs(x1)+abs(x2)+abs(x3)+abs(x4);
% find p(f(x)<1)
mc_samples=round(10.^linspace(1,4,20)); % # of Monte Carlo samples
n_repeat=5;
plist=nan(length(mc_samples),n_repeat);
for k=1:n_repeat
for i=1:length(mc_samples)
[~,p]=integrate_normal(mu,v,fun,'dom_type','fun','fun_level',1,'fun_span',3,'fun_resol',10,'mc_samples',mc_samples(i),'plotmode',0);
plist(i,k)=p;
end
end
plot(mc_samples,plist,'-k','marker','.','markersize',10)
xlabel 'Monte Carlo sample size'; ylabel 'p'
set(gca,'xscale','log')
Compute cdf and pdf of the polyhedral function
x=linspace(-1,8,50);
p=norm_fun_cdf(x,mu,v,fun);
plot(x,p,'-o')
f=norm_fun_pdf(x,mu,v,fun,'dx',0.5,'mc_samples',5e3);
plot(x,f)
Classifying 3 1d normals
normals=struct;
normals(1).mu=-1; normals(1).v=1;
normals(2).mu=1; normals(2).v=2;
normals(3).mu=4; normals(3).v=.5;
% priors and outcome values
priors=[.4 .5 .1];
vals=[4 0 0; 0 1 -2; 0 0 1];
results=classify_normals_multi(normals,'priors',priors,'vals',vals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
results = struct with fields:
norm_bd_pts: {3×1 cell}
norm_errmat: [3×3 double]
norm_err: 0.2834
norm_valmat: [3×3 double]
norm_val: 1.8466
Classifying 4 2d normals
% define struct of all normals
normals=struct;
normals(1).mu=[1;0]; normals(1).v=2*eye(2);
normals(2).mu=[0;1]; normals(2).v=eye(2);
normals(3).mu=[-1;0]; normals(3).v=eye(2);
normals(4).mu=[0;-1]; normals(4).v=eye(2);
% plot the multi-class boundary of normal 3
plot_boundary(@(n,mu,v) opt_class_multi(n,normals,3,'mu',mu,'v',v),2,'mu',normals(3).mu,'dom_type','ray_scan')
title 'boundary of normal 3'
% classify
results=classify_normals_multi(normals); axis image
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
% now classify using samples from these normals, with outcome values
samples=struct;
for i=1:4
samples(i).sample=mvnrnd(normals(i).mu,normals(i).v,1e4);
end
vals=diag([1 2 3 4]);
results_samp=classify_normals_multi(samples,'input_type','samp','vals',vals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results_samp = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.4602
norm_valmat: [4×4 double]
norm_val: 1.5604
samp_errmat: [4×4 double]
samp_err: 0.4597
samp_valmat: [4×4 double]
samp_val: 62507
axis image
% now classify using samples from t distributions similar to these normals
samples=struct;
for i=1:4
samples(i).sample=mvtrnd(normals(i).v,3,1e4)+normals(i).mu';
end
results_samp=classify_normals_multi(samples,'input_type','samp')
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results_samp = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.5760
samp_errmat: [4×4 double]
samp_err: 0.4583
axis image; axis([-6 6 -6 6])
[paper fig. 2f] Classifying 7 2d normals
% define struct of all normals
normals=struct;
normals(1).mu=[2;0]; normals(1).v=[2 1; 1 2];
normals(2).mu=[0;1]; normals(2).v=[.5 0; 0 1];
normals(3).mu=[-1;0]; normals(3).v=[1 .3; .3 1];
normals(4).mu=[0;-1]; normals(4).v=[.5 -.5; -.5 1];
normals(5).mu=[-2;2]; normals(5).v=.5*[1 1; 1 5];
normals(6).mu=[2;-3]; normals(6).v=.3*eye(2);
normals(7).mu=[-2;-2.5]; normals(7).v=[1 0; 0 .1];
% classify
results=classify_normals_multi(normals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 5
Integrating normal 1 in domain 6
Integrating normal 1 in domain 7
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 5
Integrating normal 2 in domain 6
Integrating normal 2 in domain 7
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 5
Integrating normal 3 in domain 6
Integrating normal 3 in domain 7
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 5
Integrating normal 4 in domain 6
Integrating normal 4 in domain 7
Integrating normal 5 in domain 1
Integrating normal 5 in domain 2
Integrating normal 5 in domain 3
Integrating normal 5 in domain 4
Integrating normal 5 in domain 5
Integrating normal 5 in domain 6
Integrating normal 5 in domain 7
Integrating normal 6 in domain 1
Integrating normal 6 in domain 2
Integrating normal 6 in domain 3
Integrating normal 6 in domain 4
Integrating normal 6 in domain 5
Integrating normal 6 in domain 6
Integrating normal 6 in domain 7
Integrating normal 7 in domain 1
Integrating normal 7 in domain 2
Integrating normal 7 in domain 3
Integrating normal 7 in domain 4
Integrating normal 7 in domain 5
Integrating normal 7 in domain 6
Integrating normal 7 in domain 7
results = struct with fields:
norm_bd_pts: {7×1 cell}
norm_errmat: [7×7 double]
norm_err: 0.2319
axis image; axis([-10 10 -10 10]);
Classifying 4 3d normals
normals=struct;
normals(1).mu=[1;0;0]; normals(1).v=2*eye(3);
normals(2).mu=[0;1;0]; normals(2).v=eye(3);
normals(3).mu=[-1;0;0]; normals(3).v=eye(3);
normals(4).mu=[0;-1;0]; normals(4).v=eye(3);
results=classify_normals_multi(normals)
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.4254
[paper fig. 2h] Classifying 4 4d t distribution samples
params=struct;
params(1).mu=[0;-4;-2;-2]; params(1).v=diag([1 2 3 4]);
params(2).mu=[1;1;0;0]; params(2).v=eye(4)/4;
params(3).mu=[3;4;5;7]; params(3).v=eye(4);
params(4).mu=[5;5;7;4]; params(4).v=eye(4);
priors=[.3 .5 .15 .05];
n_samp=1e3;
samples=struct;
for i=1:4
samples(i).sample=mvtrnd(params(i).v,3,n_samp)+params(i).mu';
end
results_samp=classify_normals_multi(samples,'input_type','samp','priors',priors,'mc_samples',1e3,'plotmode',[1;1;1;1])
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
results_samp = struct with fields:
norm_bd_pts: {4×1 cell}
norm_errmat: [4×4 double]
norm_err: 0.0448
samp_errmat: [4×4 double]
samp_err: 0.0675
[paper figs. 3a,b] testing the normal approximation for 2 2d classes
N_samp=1e4; n_samp=1e2;
mu_1=[5 5]; v_1=[1 .5; .5 1]; samp_1=mvnrnd(mu_1,v_1,N_samp).^3; % not normal
mu_2=[-200 -200]; v_2=[1 1; 1 3]*2e3; samp_2=mvnrnd(mu_2,v_2,N_samp);
results=classify_normals(samp_1,samp_2,'input_type','samp','samp_opt',false);
axis([-300 600 -400 600]); hold on
bd=results.norm_bd;
q0s=linspace(-100,35,100)';
true_p11=nan(size(q0s)); true_p2_2=nan(size(q0s)); % p(1|1)
norm_p11=nan(size(q0s)); norm_p2_2=nan(size(q0s));
for i=1:length(q0s)
bd_shift=bd; bd_shift.q0=q0s(i);
if ~rem(i,10)
plot_boundary(bd_shift,2,'plot_type','line','line_color',.5*[1 1 1]);
end
results_shift=classify_normals(samp_1,samp_2,'input_type','samp','dom',bd_shift,'samp_opt',false,'plotmode',false);
% true outcomes
true_p11(i)=results_shift.samp_errmat(1,1)/N_samp;
true_p2_2(i)=results_shift.samp_errmat(2,2)/N_samp;
% normal outcomes
norm_p11(i)=results_shift.norm_errmat(1,1)/.5;
norm_p2_2(i)=results_shift.norm_errmat(2,2)/.5;
end
true_p1_1_sd=sqrt(true_p11.*(1-true_p11)/n_samp);
true_p2_2_sd=sqrt(true_p2_2.*(1-true_p2_2)/n_samp);
norm_p1_1_sd=sqrt(norm_p11.*(1-norm_p11)/n_samp);
norm_p2_2_sd=sqrt(norm_p2_2.*(1-norm_p2_2)/n_samp);
figure; hold on
colors=colororder;
xline(bd.q0,'k','optimal boundary','LabelVerticalAlignment','bottom') % optimal criterion
% p(1|1)
x=q0s; y=true_p11; dy=true_p1_1_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],colors(1,:),'facealpha',.3,'linestyle','none');
y=norm_p11; dy=norm_p1_1_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',colors(1,:));
% p(2|2)
y=true_p2_2; dy=true_p2_2_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],colors(2,:),'facealpha',.3,'linestyle','none');
y=norm_p2_2; dy=norm_p2_2_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',colors(2,:));
xlim([min(q0s) max(q0s)]); ylim([-.01 1.01]); xlabel('boundary offset q_0')
[paper figs. 3c,d] testing the normal approximation for 4 4d t distributions
n_class=4; N_samp=1e4; n_samp=1e2;
% t distribution parameters
params_t=struct;
params_t(1).mu=[0;-4;-2;-2]; params_t(1).v=diag([1 2 3 4]);
params_t(2).mu=[1;1;0;0]; params_t(2).v=eye(4)/4;
params_t(3).mu=[3;4;5;7]; params_t(3).v=eye(4);
params_t(4).mu=[5;5;7;4]; params_t(4).v=eye(4);
samples=struct; params_samp=struct;
for i=1:4
% generate t samples
sample=mvtrnd(params_t(i).v,3,N_samp)+params_t(i).mu';
samples(i).sample=sample;
% mean and covariance of samples
params_samp(i).mu=mean(sample)';
params_samp(i).v=cov(sample)';
end
len_fam=10;
prior_fam=10.^linspace(-6,12,len_fam)';
vscale_fam=10.^linspace(-1,3,len_fam)';
true_p22_mean=nan(len_fam,1);
true_pe_mean=nan(len_fam,1);
true_pe_sd=nan(len_fam,1);
norm_p22_mean=nan(len_fam,1);
norm_pe_mean=nan(len_fam,1);
norm_pe_sd=nan(len_fam,1);
parfor i=1:len_fam
i
% generate family of classification domains
% uncomment when varying prior, comment when varying variance
priors_shift=[1 prior_fam(i) 1 1]; priors_shift=priors_shift/sum(priors_shift);
domains_shift=cell(n_class,1);
for k=1:n_class
domains_shift{k}=@(n,mu,v) opt_class_multi(n,params_samp,k,'mu',mu,'v',v,'priors',priors_shift);
end
% uncomment when varying variance, comment when varying prior
% params_samp_shift=params_samp;
% for k=1:n_class
% params_samp_shift(k).v=vscale_fam(i)*params_samp(k).v;
% end
% domains_shift=cell(n_class,1);
% for k=1:n_class
% domains_shift{k}=@(n,mu,v) opt_class_multi(n,params_samp_shift,k,'mu',mu,'v',v);
% end
results_shift=classify_normals_multi(samples,'input_type','samp','doms',domains_shift,'mc_samples',5e3,'plotmode',false);
% true error rates
true_p22_mean(i)=results_shift.samp_errmat(2,2)/N_samp;
true_pe_mean(i)=results_shift.samp_err;
samp_errmat_p=results_shift.samp_errmat/(n_class*N_samp);
errs=sum(~eye(length(samples)).*samp_errmat_p,2);
accs=diag(samp_errmat_p);
true_pe_sd(i)=sqrt(sum(errs.*accs)/n_samp);
% normal error rates
norm_p22_mean(i)=results_shift.norm_errmat(2,2)/sum(results_shift.norm_errmat(2,:));
norm_pe_mean(i)=results_shift.norm_err;
errs=sum(~eye(length(samples)).*results_shift.norm_errmat,2);
accs=diag(results_shift.norm_errmat);
norm_pe_sd(i)=sqrt(sum(errs.*accs)/n_samp);
end
ans =
1
ans =
2
ans =
3
ans =
4
Integrating normal 1 in domain 1
Integrating normal 1 in domain 1
Integrating normal 1 in domain 1
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
ans =
5
Integrating normal 1 in domain 1
ans =
6
Integrating normal 1 in domain 1
ans =
8
Integrating normal 1 in domain 1
ans =
10
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
ans =
7
Integrating normal 1 in domain 1
ans =
9
Integrating normal 1 in domain 1
Integrating normal 1 in domain 2
Integrating normal 1 in domain 2
Integrating normal 1 in domain 3
Integrating normal 1 in domain 3
Integrating normal 1 in domain 4
Integrating normal 1 in domain 4
Integrating normal 2 in domain 1
Integrating normal 2 in domain 1
Integrating normal 2 in domain 2
Integrating normal 2 in domain 2
Integrating normal 2 in domain 3
Integrating normal 2 in domain 3
Integrating normal 2 in domain 4
Integrating normal 2 in domain 4
Integrating normal 3 in domain 1
Integrating normal 3 in domain 1
Integrating normal 3 in domain 2
Integrating normal 3 in domain 2
Integrating normal 3 in domain 3
Integrating normal 3 in domain 3
Integrating normal 3 in domain 4
Integrating normal 3 in domain 4
Integrating normal 4 in domain 1
Integrating normal 4 in domain 1
Integrating normal 4 in domain 2
Integrating normal 4 in domain 2
Integrating normal 4 in domain 3
Integrating normal 4 in domain 3
Integrating normal 4 in domain 4
Integrating normal 4 in domain 4
true_p22_sd=sqrt(true_p22_mean.*(1-true_p22_mean)/n_samp);
norm_p22_sd=sqrt(norm_p22_mean.*(1-norm_p22_mean)/n_samp);
figure; hold on;
% uncomment when varying prior, comment when varying variance
xline(1/3,'k','optimal boundary','LabelVerticalAlignment','bottom')
xlabel('assumed prior ratio p_1/(p_2+p_3+p_4)')
x=prior_fam/3;
% uncomment when varying variance, comment when varying prior
% xline(1,'k','optimal boundary','LabelVerticalAlignment','bottom')
% xlabel('assumed variance scale')
% x=vscale_fam;
% p(2|2)
y=true_p22_mean; dy=true_p22_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],[0.9290 0.6940 0.1250],'facealpha',.3,'linestyle','none');
y=norm_p22_mean; dy=norm_p22_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',[0.9290 0.6940 0.1250]);
% pe
y=true_pe_mean; dy=true_pe_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],[0.6 0.4 0.2],'facealpha',.3,'linestyle','none');
y=norm_pe_mean; dy=norm_pe_sd;
fill([x;flipud(x)],[y-dy;flipud(y+dy)],'k','facecolor','none','edgecolor',[0.6 0.4 0.2]);
xlim([min(x) max(x)]); set(gca,'xscale','log'); ylim([-.01 1.01])
[paper fig. 5b] Actual vision research data: detecting targets on natural scenes
absent=importdata('target_absent.txt',',',1);
present=importdata('target_present.txt',',',1);
results=classify_normals(absent.data,present.data,'input_type','samp')
Iteration Func-count min f(x) Procedure
0 1 -4569
1 11 -4579 initial simplex
2 13 -4582 expand
3 14 -4582 reflect
4 15 -4582 reflect
5 16 -4582 reflect
6 17 -4582 reflect
7 19 -4583 reflect
8 20 -4583 reflect
9 21 -4583 reflect
10 23 -4585 reflect
11 24 -4585 reflect
12 25 -4585 reflect
13 27 -4586 reflect
14 28 -4586 reflect
15 29 -4586 reflect
16 41 -4586 shrink
17 42 -4586 reflect
18 44 -4586 contract inside
19 46 -4586 contract inside
20 48 -4586 contract inside
21 50 -4586 contract inside
22 52 -4586 contract inside
23 53 -4586 reflect
24 54 -4586 reflect
25 56 -4586 contract inside
26 68 -4587 shrink
27 70 -4587 contract inside
28 72 -4588 contract inside
29 73 -4588 reflect
30 74 -4588 reflect
31 86 -4588 shrink
32 98 -4588 shrink
33 99 -4588 reflect
34 100 -4588 reflect
35 101 -4588 reflect
36 102 -4588 reflect
37 114 -4588 shrink
38 116 -4588 contract outside
39 128 -4588 shrink
40 130 -4588 contract inside
41 142 -4588 shrink
42 154 -4588 shrink
43 166 -4588 shrink
44 178 -4588 shrink
45 190 -4588 shrink
46 202 -4588 shrink
47 214 -4588 shrink
48 226 -4588 shrink
49 238 -4588 shrink
50 250 -4588 shrink
51 262 -4588 shrink
52 274 -4588 shrink
53 286 -4588 shrink
54 298 -4588 shrink
55 310 -4588 shrink
56 322 -4588 shrink
57 334 -4588 shrink
58 346 -4588 shrink
59 358 -4588 shrink
60 370 -4588 shrink
61 382 -4588 shrink
62 394 -4588 shrink
63 406 -4588 shrink
64 418 -4588 shrink
65 430 -4588 shrink
66 442 -4588 shrink
67 454 -4588 shrink
68 466 -4588 shrink
69 478 -4588 shrink
70 490 -4588 shrink
71 502 -4588 shrink
72 514 -4588 shrink
73 526 -4588 shrink
74 538 -4588 shrink
75 550 -4588 shrink
76 562 -4588 shrink
77 574 -4588 shrink
78 586 -4588 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 2.173913e-04
results = struct with fields:
norm_bd: [1×1 struct]
norm_bd_pts: [3×3754 double]
norm_errmat: [2×2 double]
norm_err: 8.0400e-05
norm_dprime: 7.5475
norm_maha_dprime: 6.8175
samp_errmat: [2×2 double]
samp_err: 0.0067
samp_dprime: 4.9418
samp_dv: {[2300×1 double] [2300×1 double]}
samp_opt_bd: [1×1 struct]
samp_opt_dv: {[2300×1 double] [2300×1 double]}
samp_opt_bd_pts: [3×26070 double]
samp_opt_errmat: [2×2 double]
samp_opt_err: 0.0026
samp_opt_dprime: 5.5866
axis normal
axis([-50 170 -320 200 0 1000]); view(166,40)
xlabel('template'); ylabel('silhouette'); zlabel('edge');
[paper figs. 5d-h] Actual vision research data: detecting camouflage
load camouflage_edge_data
results_joint_2=classify_normals([edge_powers_2(:,1),edge_lpr_2(:,1)],[edge_powers_2(:,2),edge_lpr_2(:,2)],'input_type','samp');
Iteration Func-count min f(x) Procedure
0 1 -1.99656e+06
1 7 -1.99716e+06 initial simplex
2 8 -1.99716e+06 reflect
3 10 -1.99716e+06 contract inside
4 12 -1.99716e+06 contract inside
5 14 -1.99716e+06 contract outside
6 16 -1.99716e+06 contract outside
7 18 -1.99716e+06 contract inside
8 20 -1.99716e+06 contract inside
9 21 -1.99716e+06 reflect
10 23 -1.99719e+06 contract inside
11 25 -1.99756e+06 contract inside
12 27 -1.99756e+06 contract inside
13 29 -1.99758e+06 contract inside
14 31 -1.99763e+06 contract inside
15 33 -1.99774e+06 contract inside
16 34 -1.99774e+06 reflect
17 36 -1.99774e+06 contract inside
18 38 -1.99778e+06 contract inside
19 39 -1.99778e+06 reflect
20 40 -1.99778e+06 reflect
21 42 -1.99778e+06 contract inside
22 43 -1.99778e+06 reflect
23 45 -1.99784e+06 contract inside
24 47 -1.99784e+06 contract inside
25 48 -1.99784e+06 reflect
26 50 -1.99784e+06 contract inside
27 52 -1.99784e+06 contract inside
28 54 -1.99784e+06 contract inside
29 55 -1.99784e+06 reflect
30 56 -1.99784e+06 reflect
31 58 -1.99784e+06 contract inside
32 60 -1.99784e+06 contract inside
33 62 -1.99785e+06 reflect
34 64 -1.99785e+06 contract outside
35 65 -1.99785e+06 reflect
36 67 -1.99785e+06 contract inside
37 69 -1.99785e+06 contract outside
38 70 -1.99785e+06 reflect
39 72 -1.99785e+06 contract inside
40 74 -1.99785e+06 contract inside
41 76 -1.99785e+06 contract inside
42 77 -1.99785e+06 reflect
43 79 -1.99785e+06 contract inside
44 81 -1.99785e+06 reflect
45 82 -1.99785e+06 reflect
46 83 -1.99785e+06 reflect
47 84 -1.99785e+06 reflect
48 86 -1.99785e+06 contract inside
49 88 -1.99785e+06 contract outside
50 90 -1.99785e+06 reflect
51 98 -1.99785e+06 shrink
52 100 -1.99785e+06 contract outside
53 102 -1.99785e+06 contract outside
54 104 -1.99785e+06 contract inside
55 106 -1.99786e+06 contract inside
56 107 -1.99786e+06 reflect
57 115 -1.99786e+06 shrink
58 123 -1.99786e+06 shrink
59 125 -1.99786e+06 contract inside
60 127 -1.99786e+06 contract outside
61 128 -1.99786e+06 reflect
62 129 -1.99786e+06 reflect
63 137 -1.99786e+06 shrink
64 138 -1.99786e+06 reflect
65 140 -1.99786e+06 contract inside
66 142 -1.99786e+06 contract inside
67 150 -1.99786e+06 shrink
68 151 -1.99786e+06 reflect
69 152 -1.99786e+06 reflect
70 153 -1.99786e+06 reflect
71 155 -1.99786e+06 contract inside
72 163 -1.99786e+06 shrink
73 171 -1.99786e+06 shrink
74 179 -1.99786e+06 shrink
75 187 -1.99786e+06 shrink
76 195 -1.99786e+06 shrink
77 203 -1.99786e+06 shrink
78 211 -1.99786e+06 shrink
79 219 -1.99786e+06 shrink
80 227 -1.99786e+06 shrink
81 235 -1.99786e+06 shrink
82 243 -1.99786e+06 shrink
83 251 -1.99786e+06 shrink
84 259 -1.99786e+06 shrink
85 267 -1.99786e+06 shrink
86 275 -1.99786e+06 shrink
87 283 -1.99786e+06 shrink
88 291 -1.99786e+06 shrink
89 299 -1.99786e+06 shrink
90 307 -1.99786e+06 shrink
91 315 -1.99786e+06 shrink
92 323 -1.99786e+06 shrink
93 331 -1.99786e+06 shrink
94 339 -1.99786e+06 shrink
95 347 -1.99786e+06 shrink
96 355 -1.99786e+06 shrink
97 363 -1.99786e+06 shrink
98 371 -1.99786e+06 shrink
99 379 -1.99786e+06 shrink
100 387 -1.99786e+06 shrink
101 395 -1.99786e+06 shrink
102 403 -1.99786e+06 shrink
103 411 -1.99786e+06 shrink
104 419 -1.99786e+06 shrink
105 427 -1.99786e+06 shrink
106 435 -1.99786e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
axis([0.2 1 -900 400])
xlabel 'edge power'; ylabel 'edge spectrum'
results_dv_2=classify_normals(results_joint_2.samp_opt_dv{1},results_joint_2.samp_opt_dv{2},'input_type','samp','dom',@(x) x,'dom_type','fun','samp_opt',false);
axis([-50 100 0 .04])
xlabel('$q_s(${\boldmath$x$}$)$','interpreter','latex');
results_joint_4=classify_normals([edge_powers_4(:,1),edge_lpr_4(:,1)],[edge_powers_4(:,2),edge_lpr_4(:,2)],'input_type','samp','plotmode',false);
Iteration Func-count min f(x) Procedure
0 1 -1.9574e+06
1 7 -1.9574e+06 initial simplex
2 9 -1.9574e+06 contract inside
3 11 -1.9574e+06 contract outside
4 13 -1.9574e+06 contract outside
5 14 -1.9574e+06 reflect
6 16 -1.9574e+06 contract inside
7 18 -1.9574e+06 contract inside
8 20 -1.9574e+06 contract outside
9 22 -1.9574e+06 contract inside
10 24 -1.9574e+06 contract inside
11 26 -1.9574e+06 contract inside
12 28 -1.95741e+06 contract inside
13 30 -1.95782e+06 contract inside
14 32 -1.95782e+06 contract outside
15 34 -1.95783e+06 contract inside
16 36 -1.95808e+06 contract inside
17 37 -1.95808e+06 reflect
18 39 -1.95808e+06 contract inside
19 41 -1.95808e+06 contract inside
20 43 -1.95808e+06 contract inside
21 45 -1.95808e+06 contract inside
22 47 -1.95808e+06 contract inside
23 49 -1.95811e+06 contract inside
24 50 -1.95811e+06 reflect
25 52 -1.95811e+06 contract inside
26 54 -1.95811e+06 contract inside
27 56 -1.95814e+06 contract inside
28 58 -1.95818e+06 contract inside
29 60 -1.95819e+06 reflect
30 62 -1.95819e+06 contract inside
31 64 -1.95819e+06 contract inside
32 66 -1.95819e+06 contract inside
33 68 -1.95819e+06 contract inside
34 70 -1.9582e+06 contract inside
35 72 -1.9582e+06 contract inside
36 74 -1.9582e+06 contract inside
37 75 -1.9582e+06 reflect
38 77 -1.95822e+06 contract inside
39 78 -1.95822e+06 reflect
40 80 -1.95822e+06 contract inside
41 82 -1.95822e+06 contract inside
42 84 -1.95822e+06 contract inside
43 86 -1.95822e+06 contract inside
44 88 -1.95822e+06 contract inside
45 89 -1.95822e+06 reflect
46 90 -1.95822e+06 reflect
47 92 -1.95823e+06 expand
48 93 -1.95823e+06 reflect
49 95 -1.95823e+06 contract inside
50 96 -1.95823e+06 reflect
51 97 -1.95823e+06 reflect
52 99 -1.95824e+06 reflect
53 101 -1.95824e+06 contract inside
54 103 -1.95825e+06 expand
55 104 -1.95825e+06 reflect
56 106 -1.95825e+06 reflect
57 107 -1.95825e+06 reflect
58 115 -1.95825e+06 shrink
59 117 -1.95825e+06 contract inside
60 118 -1.95825e+06 reflect
61 120 -1.95825e+06 contract inside
62 122 -1.95826e+06 expand
63 123 -1.95826e+06 reflect
64 125 -1.95826e+06 contract inside
65 127 -1.95826e+06 contract inside
66 128 -1.95826e+06 reflect
67 130 -1.95827e+06 expand
68 132 -1.95827e+06 contract outside
69 133 -1.95827e+06 reflect
70 134 -1.95827e+06 reflect
71 135 -1.95827e+06 reflect
72 137 -1.95828e+06 expand
73 138 -1.95828e+06 reflect
74 140 -1.95828e+06 reflect
75 142 -1.95829e+06 expand
76 143 -1.95829e+06 reflect
77 145 -1.95829e+06 contract inside
78 147 -1.95834e+06 expand
79 148 -1.95834e+06 reflect
80 149 -1.95834e+06 reflect
81 150 -1.95834e+06 reflect
82 151 -1.95834e+06 reflect
83 153 -1.95838e+06 expand
84 154 -1.95838e+06 reflect
85 155 -1.95838e+06 reflect
86 156 -1.95838e+06 reflect
87 158 -1.9584e+06 expand
88 159 -1.9584e+06 reflect
89 161 -1.95846e+06 expand
90 162 -1.95846e+06 reflect
91 163 -1.95846e+06 reflect
92 164 -1.95846e+06 reflect
93 166 -1.9585e+06 expand
94 167 -1.9585e+06 reflect
95 169 -1.95856e+06 expand
96 170 -1.95856e+06 reflect
97 172 -1.95862e+06 expand
98 173 -1.95862e+06 reflect
99 174 -1.95862e+06 reflect
100 175 -1.95862e+06 reflect
101 177 -1.95865e+06 expand
102 178 -1.95865e+06 reflect
103 179 -1.95865e+06 reflect
104 180 -1.95865e+06 reflect
105 182 -1.95867e+06 expand
106 184 -1.95867e+06 contract outside
107 186 -1.95872e+06 expand
108 188 -1.95872e+06 contract inside
109 189 -1.95872e+06 reflect
110 191 -1.95872e+06 contract inside
111 192 -1.95872e+06 reflect
112 194 -1.95878e+06 expand
113 195 -1.95878e+06 reflect
114 196 -1.95878e+06 reflect
115 197 -1.95878e+06 reflect
116 199 -1.95888e+06 expand
117 200 -1.95888e+06 reflect
118 201 -1.95888e+06 reflect
119 203 -1.95898e+06 expand
120 204 -1.95898e+06 reflect
121 205 -1.95898e+06 reflect
122 207 -1.9591e+06 expand
123 208 -1.9591e+06 reflect
124 209 -1.9591e+06 reflect
125 211 -1.95917e+06 expand
126 213 -1.95933e+06 expand
127 214 -1.95933e+06 reflect
128 215 -1.95933e+06 reflect
129 217 -1.95945e+06 expand
130 218 -1.95945e+06 reflect
131 219 -1.95945e+06 reflect
132 221 -1.95961e+06 expand
133 222 -1.95961e+06 reflect
134 223 -1.95961e+06 reflect
135 225 -1.95961e+06 contract inside
136 227 -1.95964e+06 reflect
137 229 -1.95969e+06 reflect
138 230 -1.95969e+06 reflect
139 232 -1.95969e+06 contract inside
140 234 -1.95969e+06 contract inside
141 235 -1.95969e+06 reflect
142 236 -1.95969e+06 reflect
143 237 -1.95969e+06 reflect
144 238 -1.95969e+06 reflect
145 240 -1.95969e+06 contract outside
146 242 -1.95969e+06 contract inside
147 244 -1.95971e+06 reflect
148 246 -1.95976e+06 expand
149 248 -1.95976e+06 contract outside
150 249 -1.95976e+06 reflect
151 250 -1.95976e+06 reflect
152 252 -1.95981e+06 expand
153 253 -1.95981e+06 reflect
154 254 -1.95981e+06 reflect
155 256 -1.95996e+06 expand
156 257 -1.95996e+06 reflect
157 259 -1.96005e+06 expand
158 260 -1.96005e+06 reflect
159 262 -1.96018e+06 expand
160 263 -1.96018e+06 reflect
161 265 -1.96059e+06 expand
162 266 -1.96059e+06 reflect
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165 270 -1.96078e+06 expand
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168 275 -1.96135e+06 reflect
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171 279 -1.96135e+06 contract inside
172 280 -1.96135e+06 reflect
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175 285 -1.96172e+06 expand
176 287 -1.96179e+06 reflect
177 288 -1.96179e+06 reflect
178 290 -1.96179e+06 contract inside
179 291 -1.96179e+06 reflect
180 293 -1.96179e+06 contract inside
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182 296 -1.96179e+06 reflect
183 298 -1.96179e+06 contract inside
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186 304 -1.96188e+06 reflect
187 306 -1.96188e+06 contract inside
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190 311 -1.9619e+06 reflect
191 313 -1.9619e+06 contract outside
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195 319 -1.96195e+06 reflect
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198 324 -1.96195e+06 reflect
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203 331 -1.96195e+06 reflect
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206 336 -1.96195e+06 reflect
207 338 -1.96195e+06 contract inside
208 339 -1.96195e+06 reflect
209 341 -1.96195e+06 contract inside
210 342 -1.96195e+06 reflect
211 350 -1.96196e+06 shrink
212 351 -1.96196e+06 reflect
213 353 -1.96196e+06 contract outside
214 354 -1.96196e+06 reflect
215 356 -1.96196e+06 contract inside
216 358 -1.96196e+06 contract outside
217 359 -1.96196e+06 reflect
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219 362 -1.96196e+06 contract inside
220 364 -1.96196e+06 contract inside
221 365 -1.96196e+06 reflect
222 367 -1.96196e+06 contract inside
223 368 -1.96196e+06 reflect
224 376 -1.96196e+06 shrink
225 377 -1.96196e+06 reflect
226 379 -1.96196e+06 contract outside
227 381 -1.96196e+06 contract inside
228 389 -1.96197e+06 shrink
229 391 -1.96197e+06 contract inside
230 392 -1.96197e+06 reflect
231 393 -1.96197e+06 reflect
232 395 -1.96197e+06 contract outside
233 396 -1.96197e+06 reflect
234 398 -1.96197e+06 contract inside
235 399 -1.96197e+06 reflect
236 400 -1.96197e+06 reflect
237 401 -1.96197e+06 reflect
238 409 -1.96197e+06 shrink
239 411 -1.96197e+06 contract inside
240 412 -1.96197e+06 reflect
241 414 -1.96197e+06 contract inside
242 415 -1.96197e+06 reflect
243 423 -1.96197e+06 shrink
244 424 -1.96197e+06 reflect
245 426 -1.96197e+06 contract outside
246 434 -1.96197e+06 shrink
247 442 -1.96197e+06 shrink
248 444 -1.96197e+06 contract inside
249 446 -1.96197e+06 contract inside
250 454 -1.96197e+06 shrink
251 462 -1.96197e+06 shrink
252 470 -1.96197e+06 shrink
253 478 -1.96197e+06 shrink
254 486 -1.96197e+06 shrink
255 494 -1.96197e+06 shrink
256 502 -1.96197e+06 shrink
257 510 -1.96197e+06 shrink
258 518 -1.96197e+06 shrink
259 526 -1.96197e+06 shrink
260 534 -1.96197e+06 shrink
261 542 -1.96197e+06 shrink
262 550 -1.96197e+06 shrink
263 558 -1.96197e+06 shrink
264 566 -1.96197e+06 shrink
265 574 -1.96197e+06 shrink
266 582 -1.96197e+06 shrink
267 590 -1.96197e+06 shrink
268 598 -1.96197e+06 shrink
269 606 -1.96197e+06 shrink
270 614 -1.96197e+06 shrink
271 622 -1.96197e+06 shrink
272 630 -1.96197e+06 shrink
273 638 -1.96197e+06 shrink
274 646 -1.96197e+06 shrink
275 654 -1.96197e+06 shrink
276 662 -1.96197e+06 shrink
277 670 -1.96197e+06 shrink
278 678 -1.96197e+06 shrink
279 686 -1.96197e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
results_joint_8=classify_normals([edge_powers_8(:,1),edge_lpr_8(:,1)],[edge_powers_8(:,2),edge_lpr_8(:,2)],'input_type','samp','plotmode',false);
Iteration Func-count min f(x) Procedure
0 1 -1.78592e+06
1 7 -1.78592e+06 initial simplex
2 9 -1.78592e+06 contract inside
3 11 -1.78592e+06 contract inside
4 13 -1.78592e+06 contract inside
5 15 -1.78699e+06 contract outside
6 17 -1.78713e+06 contract inside
7 19 -1.78734e+06 contract outside
8 21 -1.78746e+06 contract inside
9 23 -1.78777e+06 contract inside
10 25 -1.78777e+06 contract inside
11 27 -1.7878e+06 contract inside
12 29 -1.78788e+06 contract outside
13 31 -1.78788e+06 contract outside
14 33 -1.78792e+06 contract inside
15 35 -1.78837e+06 reflect
16 36 -1.78837e+06 reflect
17 38 -1.78837e+06 contract inside
18 39 -1.78837e+06 reflect
19 41 -1.78837e+06 contract inside
20 43 -1.78837e+06 contract inside
21 44 -1.78837e+06 reflect
22 46 -1.78893e+06 expand
23 47 -1.78893e+06 reflect
24 48 -1.78893e+06 reflect
25 49 -1.78893e+06 reflect
26 50 -1.78893e+06 reflect
27 51 -1.78893e+06 reflect
28 53 -1.78903e+06 expand
29 54 -1.78903e+06 reflect
30 56 -1.78903e+06 contract inside
31 58 -1.78924e+06 expand
32 59 -1.78924e+06 reflect
33 61 -1.7893e+06 reflect
34 63 -1.78992e+06 expand
35 64 -1.78992e+06 reflect
36 66 -1.78992e+06 contract inside
37 67 -1.78992e+06 reflect
38 69 -1.78996e+06 expand
39 71 -1.78999e+06 reflect
40 73 -1.79056e+06 expand
41 74 -1.79056e+06 reflect
42 76 -1.79157e+06 expand
43 77 -1.79157e+06 reflect
44 78 -1.79157e+06 reflect
45 79 -1.79157e+06 reflect
46 81 -1.79183e+06 reflect
47 83 -1.79314e+06 expand
48 84 -1.79314e+06 reflect
49 85 -1.79314e+06 reflect
50 87 -1.79341e+06 expand
51 89 -1.79449e+06 expand
52 91 -1.79567e+06 expand
53 92 -1.79567e+06 reflect
54 93 -1.79567e+06 reflect
55 95 -1.79608e+06 expand
56 97 -1.79781e+06 expand
57 99 -1.79889e+06 expand
58 101 -1.79889e+06 contract inside
59 102 -1.79889e+06 reflect
60 103 -1.79889e+06 reflect
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62 105 -1.79889e+06 reflect
63 107 -1.79889e+06 contract inside
64 108 -1.79889e+06 reflect
65 110 -1.79907e+06 reflect
66 112 -1.79907e+06 contract outside
67 114 -1.79907e+06 contract inside
68 115 -1.79907e+06 reflect
69 116 -1.79907e+06 reflect
70 118 -1.79914e+06 contract inside
71 120 -1.79914e+06 contract inside
72 122 -1.79927e+06 reflect
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74 125 -1.79927e+06 contract inside
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76 128 -1.79927e+06 reflect
77 130 -1.79927e+06 contract inside
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81 137 -1.79928e+06 reflect
82 139 -1.79928e+06 contract inside
83 140 -1.79928e+06 reflect
84 142 -1.79928e+06 contract outside
85 144 -1.79928e+06 contract inside
86 145 -1.79928e+06 reflect
87 146 -1.79928e+06 reflect
88 148 -1.79928e+06 contract inside
89 149 -1.79928e+06 reflect
90 150 -1.79928e+06 reflect
91 158 -1.79928e+06 shrink
92 160 -1.79928e+06 contract outside
93 161 -1.79928e+06 reflect
94 162 -1.79928e+06 reflect
95 164 -1.79931e+06 expand
96 166 -1.79931e+06 contract inside
97 168 -1.79931e+06 contract inside
98 176 -1.79931e+06 shrink
99 177 -1.79931e+06 reflect
100 178 -1.79931e+06 reflect
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103 182 -1.79931e+06 contract inside
104 190 -1.79932e+06 shrink
105 191 -1.79932e+06 reflect
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107 194 -1.79932e+06 contract inside
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111 208 -1.79932e+06 shrink
112 210 -1.79932e+06 contract outside
113 212 -1.79932e+06 contract inside
114 213 -1.79932e+06 reflect
115 215 -1.79933e+06 reflect
116 217 -1.79933e+06 contract inside
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118 227 -1.79933e+06 shrink
119 229 -1.79933e+06 contract inside
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122 234 -1.79933e+06 reflect
123 236 -1.79933e+06 contract inside
124 237 -1.79933e+06 reflect
125 239 -1.79933e+06 contract inside
126 240 -1.79933e+06 reflect
127 248 -1.79933e+06 shrink
128 249 -1.79933e+06 reflect
129 250 -1.79933e+06 reflect
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131 252 -1.79933e+06 reflect
132 254 -1.79933e+06 contract inside
133 255 -1.79933e+06 reflect
134 257 -1.79933e+06 contract outside
135 265 -1.79933e+06 shrink
136 267 -1.79933e+06 contract outside
137 268 -1.79933e+06 reflect
138 270 -1.79933e+06 contract inside
139 272 -1.79933e+06 contract inside
140 274 -1.79933e+06 contract inside
141 282 -1.79933e+06 shrink
142 284 -1.79933e+06 contract inside
143 286 -1.79933e+06 contract inside
144 288 -1.79933e+06 contract inside
145 296 -1.79933e+06 shrink
146 298 -1.79933e+06 contract inside
147 306 -1.79933e+06 shrink
148 314 -1.79933e+06 shrink
149 322 -1.79933e+06 shrink
150 330 -1.79933e+06 shrink
151 338 -1.79933e+06 shrink
152 346 -1.79933e+06 shrink
153 354 -1.79933e+06 shrink
154 362 -1.79933e+06 shrink
155 363 -1.79933e+06 reflect
156 365 -1.79933e+06 contract outside
157 373 -1.79933e+06 shrink
158 381 -1.79933e+06 shrink
159 389 -1.79933e+06 shrink
160 397 -1.79933e+06 shrink
161 405 -1.79933e+06 shrink
162 413 -1.79933e+06 shrink
163 421 -1.79933e+06 shrink
164 429 -1.79933e+06 shrink
165 437 -1.79933e+06 shrink
166 445 -1.79933e+06 shrink
167 453 -1.79933e+06 shrink
168 461 -1.79933e+06 shrink
169 469 -1.79933e+06 shrink
170 477 -1.79933e+06 shrink
171 485 -1.79933e+06 shrink
172 493 -1.79933e+06 shrink
173 501 -1.79933e+06 shrink
174 509 -1.79933e+06 shrink
175 517 -1.79933e+06 shrink
176 525 -1.79933e+06 shrink
177 533 -1.79933e+06 shrink
178 541 -1.79933e+06 shrink
179 549 -1.79933e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
results_dv_joint=classify_normals([results_joint_2.samp_opt_dv{1},results_joint_4.samp_opt_dv{1},results_joint_8.samp_opt_dv{1}],[results_joint_2.samp_opt_dv{2},results_joint_4.samp_opt_dv{2},results_joint_8.samp_opt_dv{2}],'input_type','samp');
Iteration Func-count min f(x) Procedure
0 1 -1.99145e+06
1 11 -1.9929e+06 initial simplex
2 12 -1.9929e+06 reflect
3 13 -1.9929e+06 reflect
4 14 -1.9929e+06 reflect
5 15 -1.9929e+06 reflect
6 16 -1.9929e+06 reflect
7 17 -1.9929e+06 reflect
8 19 -1.99367e+06 expand
9 20 -1.99367e+06 reflect
10 22 -1.99445e+06 expand
11 23 -1.99445e+06 reflect
12 24 -1.99445e+06 reflect
13 25 -1.99445e+06 reflect
14 26 -1.99445e+06 reflect
15 27 -1.99445e+06 reflect
16 29 -1.99523e+06 expand
17 30 -1.99523e+06 reflect
18 32 -1.99604e+06 expand
19 33 -1.99604e+06 reflect
20 34 -1.99604e+06 reflect
21 35 -1.99604e+06 reflect
22 36 -1.99604e+06 reflect
23 37 -1.99604e+06 reflect
24 39 -1.99682e+06 expand
25 40 -1.99682e+06 reflect
26 41 -1.99682e+06 reflect
27 42 -1.99682e+06 reflect
28 43 -1.99682e+06 reflect
29 45 -1.99721e+06 expand
30 46 -1.99721e+06 reflect
31 47 -1.99721e+06 reflect
32 48 -1.99721e+06 reflect
33 49 -1.99721e+06 reflect
34 50 -1.99721e+06 reflect
35 51 -1.99721e+06 reflect
36 53 -1.99734e+06 expand
37 54 -1.99734e+06 reflect
38 55 -1.99734e+06 reflect
39 57 -1.99782e+06 expand
40 58 -1.99782e+06 reflect
41 59 -1.99782e+06 reflect
42 60 -1.99782e+06 reflect
43 61 -1.99782e+06 reflect
44 63 -1.99782e+06 contract outside
45 64 -1.99782e+06 reflect
46 65 -1.99782e+06 reflect
47 67 -1.99796e+06 expand
48 69 -1.99796e+06 contract inside
49 70 -1.99796e+06 reflect
50 71 -1.99796e+06 reflect
51 72 -1.99796e+06 reflect
52 74 -1.99806e+06 expand
53 75 -1.99806e+06 reflect
54 76 -1.99806e+06 reflect
55 78 -1.99806e+06 reflect
56 80 -1.99809e+06 expand
57 81 -1.99809e+06 reflect
58 82 -1.99809e+06 reflect
59 84 -1.9981e+06 reflect
60 85 -1.9981e+06 reflect
61 86 -1.9981e+06 reflect
62 87 -1.9981e+06 reflect
63 89 -1.9981e+06 contract inside
64 91 -1.9981e+06 contract inside
65 93 -1.9981e+06 contract inside
66 94 -1.9981e+06 reflect
67 96 -1.9981e+06 contract inside
68 98 -1.9981e+06 contract inside
69 99 -1.9981e+06 reflect
70 101 -1.9981e+06 contract inside
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72 104 -1.9981e+06 reflect
73 106 -1.9981e+06 contract inside
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75 110 -1.9981e+06 reflect
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77 113 -1.9981e+06 contract inside
78 115 -1.9981e+06 contract inside
79 116 -1.9981e+06 reflect
80 118 -1.99811e+06 reflect
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82 120 -1.99811e+06 reflect
83 121 -1.99811e+06 reflect
84 123 -1.99811e+06 contract inside
85 125 -1.99811e+06 contract outside
86 126 -1.99811e+06 reflect
87 127 -1.99811e+06 reflect
88 128 -1.99811e+06 reflect
89 130 -1.99811e+06 contract inside
90 132 -1.99812e+06 expand
91 134 -1.99812e+06 contract outside
92 136 -1.99812e+06 contract inside
93 138 -1.99812e+06 contract inside
94 139 -1.99812e+06 reflect
95 140 -1.99812e+06 reflect
96 142 -1.99812e+06 contract inside
97 144 -1.99812e+06 contract inside
98 145 -1.99812e+06 reflect
99 146 -1.99812e+06 reflect
100 158 -1.99812e+06 shrink
101 160 -1.99812e+06 contract outside
102 161 -1.99812e+06 reflect
103 162 -1.99812e+06 reflect
104 164 -1.99812e+06 contract outside
105 166 -1.99812e+06 contract inside
106 168 -1.99812e+06 contract inside
107 170 -1.99812e+06 contract outside
108 171 -1.99812e+06 reflect
109 172 -1.99812e+06 reflect
110 174 -1.99812e+06 contract inside
111 176 -1.99812e+06 contract inside
112 178 -1.99812e+06 contract outside
113 180 -1.99812e+06 contract inside
114 181 -1.99812e+06 reflect
115 193 -1.99812e+06 shrink
116 194 -1.99812e+06 reflect
117 195 -1.99812e+06 reflect
118 196 -1.99812e+06 reflect
119 208 -1.99812e+06 shrink
120 210 -1.99812e+06 contract outside
121 212 -1.99812e+06 contract outside
122 213 -1.99812e+06 reflect
123 215 -1.99812e+06 contract outside
124 227 -1.99812e+06 shrink
125 228 -1.99812e+06 reflect
126 229 -1.99812e+06 reflect
127 230 -1.99812e+06 reflect
128 231 -1.99812e+06 reflect
129 232 -1.99812e+06 reflect
130 233 -1.99812e+06 reflect
131 235 -1.99812e+06 contract outside
132 247 -1.99812e+06 shrink
133 248 -1.99812e+06 reflect
134 260 -1.99812e+06 shrink
135 272 -1.99812e+06 shrink
136 273 -1.99812e+06 reflect
137 285 -1.99812e+06 shrink
138 287 -1.99812e+06 contract inside
139 299 -1.99812e+06 shrink
140 311 -1.99812e+06 shrink
141 323 -1.99812e+06 shrink
142 335 -1.99812e+06 shrink
143 347 -1.99812e+06 shrink
144 359 -1.99812e+06 shrink
145 371 -1.99812e+06 shrink
146 383 -1.99812e+06 shrink
147 395 -1.99812e+06 shrink
148 407 -1.99812e+06 shrink
149 419 -1.99812e+06 shrink
150 431 -1.99812e+06 shrink
151 443 -1.99812e+06 shrink
152 455 -1.99812e+06 shrink
153 467 -1.99812e+06 shrink
154 479 -1.99812e+06 shrink
155 491 -1.99812e+06 shrink
156 503 -1.99812e+06 shrink
157 515 -1.99812e+06 shrink
158 527 -1.99812e+06 shrink
159 539 -1.99812e+06 shrink
160 551 -1.99812e+06 shrink
161 563 -1.99812e+06 shrink
162 575 -1.99812e+06 shrink
163 587 -1.99812e+06 shrink
164 599 -1.99812e+06 shrink
165 611 -1.99812e+06 shrink
166 623 -1.99812e+06 shrink
167 635 -1.99812e+06 shrink
168 647 -1.99812e+06 shrink
169 659 -1.99812e+06 shrink
170 671 -1.99812e+06 shrink
171 683 -1.99812e+06 shrink
172 695 -1.99812e+06 shrink
173 707 -1.99812e+06 shrink
174 719 -1.99812e+06 shrink
175 731 -1.99812e+06 shrink
176 743 -1.99812e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
axis image; axis([-100 125 -50 75 -50 50]); view(-20,24)
xlabel('$q_s(${\boldmath$x$}$)$ (2px)','interpreter','latex');
ylabel('$q_s(${\boldmath$x$}$)$ (4px)','interpreter','latex');
zlabel('$q_s(${\boldmath$x$}$)$ (8px)','interpreter','latex');
results_all=classify_normals([edge_powers_2(:,1),edge_lpr_2(:,1),...
edge_powers_4(:,1),edge_lpr_4(:,1),...
edge_powers_8(:,1),edge_lpr_8(:,1)],...
[edge_powers_2(:,2),edge_lpr_2(:,2),...
edge_powers_4(:,2),edge_lpr_4(:,2),...
edge_powers_8(:,2),edge_lpr_8(:,2)],'input_type','samp');
Iteration Func-count min f(x) Procedure
0 1 -1.99471e+06
1 29 -1.99689e+06 initial simplex
2 30 -1.99689e+06 reflect
3 31 -1.99689e+06 reflect
4 32 -1.99689e+06 reflect
5 34 -1.99689e+06 contract inside
6 36 -1.99689e+06 contract inside
7 38 -1.99689e+06 contract inside
8 40 -1.99689e+06 contract inside
9 41 -1.99689e+06 reflect
10 43 -1.99689e+06 contract inside
11 45 -1.99689e+06 contract inside
12 46 -1.99689e+06 reflect
13 48 -1.99689e+06 contract inside
14 50 -1.99689e+06 contract inside
15 52 -1.99689e+06 contract inside
16 54 -1.99691e+06 contract inside
17 56 -1.99718e+06 reflect
18 57 -1.99718e+06 reflect
19 59 -1.99721e+06 contract inside
20 61 -1.99749e+06 reflect
21 63 -1.99749e+06 contract inside
22 64 -1.99749e+06 reflect
23 65 -1.99749e+06 reflect
24 66 -1.99749e+06 reflect
25 67 -1.99749e+06 reflect
26 68 -1.99749e+06 reflect
27 69 -1.99749e+06 reflect
28 70 -1.99749e+06 reflect
29 72 -1.99749e+06 contract inside
30 74 -1.99749e+06 contract inside
31 76 -1.99749e+06 contract inside
32 78 -1.99749e+06 contract inside
33 80 -1.99749e+06 contract inside
34 82 -1.99749e+06 contract inside
35 84 -1.99749e+06 contract inside
36 86 -1.99749e+06 contract inside
37 88 -1.99749e+06 contract inside
38 90 -1.99749e+06 contract inside
39 91 -1.99749e+06 reflect
40 93 -1.99749e+06 contract outside
41 94 -1.99749e+06 reflect
42 96 -1.99749e+06 contract inside
43 98 -1.99749e+06 contract inside
44 100 -1.99749e+06 contract inside
45 101 -1.99749e+06 reflect
46 102 -1.99749e+06 reflect
47 103 -1.99749e+06 reflect
48 104 -1.99749e+06 reflect
49 106 -1.99749e+06 contract inside
50 108 -1.99749e+06 contract inside
51 110 -1.99749e+06 contract inside
52 112 -1.99749e+06 contract inside
53 114 -1.99749e+06 contract inside
54 116 -1.99749e+06 contract inside
55 118 -1.99749e+06 contract inside
56 119 -1.99749e+06 reflect
57 121 -1.99749e+06 contract inside
58 122 -1.99749e+06 reflect
59 124 -1.99749e+06 contract outside
60 126 -1.99749e+06 contract inside
61 128 -1.99752e+06 contract inside
62 130 -1.99752e+06 contract inside
63 132 -1.99752e+06 contract inside
64 133 -1.99752e+06 reflect
65 135 -1.99752e+06 contract outside
66 137 -1.99752e+06 contract inside
67 139 -1.99752e+06 contract inside
68 141 -1.99752e+06 contract inside
69 143 -1.99754e+06 contract inside
70 145 -1.99757e+06 contract inside
71 146 -1.99757e+06 reflect
72 148 -1.99757e+06 contract outside
73 150 -1.99757e+06 contract outside
74 151 -1.99757e+06 reflect
75 152 -1.99757e+06 reflect
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684 900 -1.99833e+06 reflect
685 902 -1.99833e+06 contract inside
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706 965 -1.99833e+06 shrink
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729 1027 -1.99833e+06 shrink
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743 1075 -1.99833e+06 shrink
744 1076 -1.99833e+06 reflect
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746 1108 -1.99833e+06 shrink
747 1109 -1.99833e+06 reflect
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750 1112 -1.99833e+06 reflect
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753 1144 -1.99833e+06 shrink
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756 1177 -1.99833e+06 contract inside
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758 1180 -1.99833e+06 reflect
759 1182 -1.99833e+06 contract outside
760 1212 -1.99833e+06 shrink
761 1242 -1.99833e+06 shrink
762 1272 -1.99833e+06 shrink
763 1302 -1.99833e+06 shrink
764 1332 -1.99833e+06 shrink
765 1362 -1.99833e+06 shrink
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767 1422 -1.99833e+06 shrink
768 1452 -1.99833e+06 shrink
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780 1812 -1.99833e+06 shrink
781 1842 -1.99833e+06 shrink
782 1872 -1.99833e+06 shrink
783 1902 -1.99833e+06 shrink
784 1932 -1.99833e+06 shrink
785 1962 -1.99833e+06 shrink
786 1992 -1.99833e+06 shrink
787 2022 -1.99833e+06 shrink
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790 2112 -1.99833e+06 shrink
791 2142 -1.99833e+06 shrink
792 2172 -1.99833e+06 shrink
793 2202 -1.99833e+06 shrink
794 2232 -1.99833e+06 shrink
Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 0.000000e+00
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 5.000000e-07
results_all_samp_opt=classify_normals([edge_powers_2(:,1),edge_lpr_2(:,1),...
edge_powers_4(:,1),edge_lpr_4(:,1),...
edge_powers_8(:,1),edge_lpr_8(:,1)],...
[edge_powers_2(:,2),edge_lpr_2(:,2),...
edge_powers_4(:,2),edge_lpr_4(:,2),...
edge_powers_8(:,2),edge_lpr_8(:,2)],'input_type','samp',...
'dom',results_all.samp_opt_bd,'samp_opt',false);
xlabel('$q_s(${\boldmath$x$}$)$ (2, 4 and 8px)','interpreter','latex');
axis([-125 150 0 .015])
